Polynomial-Time Pseudodeterministic Construction of Primes

TL;DR

This paper presents an unconditional polynomial-time randomized algorithm that can construct prime numbers in polynomial time for infinitely many input lengths, solving a long-standing open problem in pseudodeterministic algorithms.

Contribution

It provides the first unconditional polynomial-time pseudodeterministic algorithm for infinitely-often prime construction, advancing the understanding of derandomization and pseudodeterminism.

Findings

Constructs primes in polynomial time infinitely often

Generalizes to dense polynomial-time decidable properties

Improves upon previous subexponential constructions

Abstract

A randomized algorithm for a search problem is *pseudodeterministic* if it produces a fixed canonical solution to the search problem with high probability. In their seminal work on the topic, Gat and Goldwasser posed as their main open problem whether prime numbers can be pseudodeterministically constructed in polynomial time. We provide a positive solution to this question in the infinitely-often regime. In more detail, we give an *unconditional* polynomial-time randomized algorithm $B$ such that, for infinitely many values of $n$, $B(1^n)$ outputs a canonical $n$-bit prime $p_n$ with high probability. More generally, we prove that for every dense property $Q$ of strings that can be decided in polynomial time, there is an infinitely-often pseudodeterministic polynomial-time construction of strings satisfying $Q$. This improves upon a subexponential-time construction of Oliveira and Santhanam. Our construction uses several new ideas, including a novel bootstrapping technique for pseudodeterministic constructions, and a quantitative optimization of the uniform hardness-randomness framework of Chen and Tell, using a variant of the Shaltiel--Umans generator.